The sequence $$x _{1},x _{2},x _{3},...$$is defined by $$x _{k}=1/(k ^{2}+k)$$. A sum of consecutive terms $$x _{m}, x _{m+1}, +...+x _{n}=1/29$$. Find n and m.

Question

The sequence $$x _{1},x _{2},x _{3},...$$is defined by  $$x _{k}=1/(k ^{2}+k)$$. A sum of consecutive terms $$x _{m}, x _{m+1}, +...+x _{n}=1/29$$. Find n and m.

mr.math · Answer

I've got this equation for m and n:
$$m=\frac{29(n+1)}{n+30}$$
We need to find positive integer solutions for n and m.

mr.math · Answer

Obviously $n>m$.

jamesj · Answer

Notice that
$$ \frac{1}{k} - \frac{1}{k+1} = \frac{1}{k(k+1)} $$
Hence
$$ 1 - \frac{1}{n+1} = \sum_{k=1}^n \frac{1}{k(k+1)} =: S_n$$

Now, as 29 is prime, this would seem to suggest we need to write
$$ \frac{1}{29} = 1 - \left(1 - \frac{1}{28+1}\right)  = S_\infty - S_{28} $$